Cleaning would be my first port of call. You should be able to do it without pulling much apart (modern cases tend come with the ability to easily open them).

Redoing the graphics cooling is normally quite hard owing to the custom designs used by different companies.
Before you get to that though check the thermal paste (if any) used to bridge the chips and the heatsink, over the last few months I have both experienced myself, had others experience it and read about very bad jobs applying it or really low grade paste used (margins are thinner than ever and silicon "works").

Check fan speeds, some drivers/BIOS settings have the ability to change these and often will plump for the "quieter" modes at the cost of heat dissipation. Crank it up and just up the volume on your machine or wear headphones if necessary.

Reshuffling some cards/wires to allow more airflow is a good idea. Likewise improving airflow to the case (do not bury it right in the corner or a the desk/wall) is also good.

New cards have need for extra power connectors (with cooling being a part of them), check your PSU is up to the job.

If you have a case fan directing it help out the graphics card is an idea.

If all else fails try underclocking the card. It will reduce heat buildup, often quite substantially; heat dissipation is exponential in nature so a small drop in some things can lead to a bit drop in heat, timings/speeds will be set so as to allow the maximum with as small a safety margin as possible.
Sidenote if it is the ram overheating you could increase the GPU speed to compensate (or vice versa), this can help to offset the performance reduction.
Similarly you can try using the low quality options of the drivers (less work means less heat), game speeds will increase and if you do it right it will not look all that different (and if you do it really well you can even increase draw distance which is far nicer than a bit more anti-aliasing).

If you have to upgrade the cooling (and sticking another fan in via the motherboard/conventional methods is not possible) first look to the manufacturer to see if they have such an option.

Secondly fans/motors used in computers are simple devices despite what the companies may have you believe (the day I see a squirrel cage motor in a computer is the day I retract that statement) so you can tweak existing designs to work with them. You can also overvoltage a fan which should get it spinning faster.